Solve Nested Radicals: Cube Root of Square Root of 36

Complete the following exercise:

$\sqrt[3]{\sqrt{36}}=$

To solve this problem, let's analyze and simplify the given expression $\sqrt[3]{\sqrt{36}}$.

• Step 1: Identify the root operations. We have a square root, $\sqrt{36}$, and a cube root, $\sqrt[3]{\ldots}$.

• Step 2: Use the formula for roots for a root of a root: $\sqrt[m]{\sqrt[n]{x}} = x^{\frac{1}{mn}}$.

• Step 3: Apply this formula to the problem. In this case, the first operation is a square root, which can be written as $36^{\frac{1}{2}}$, and the second operation is a cube root. Therefore, $\sqrt[3]{\sqrt{36}} = (36^{\frac{1}{2}})^{\frac{1}{3}}$.

• Step 4: Simplify using the power of a power rule, which allows us to multiply exponents: $(36^{\frac{1}{2}})^{\frac{1}{3}} = 36^{\frac{1}{2 \times 3}} = 36^{\frac{1}{6}}$.

Thus, the expression $\sqrt[3]{\sqrt{36}}$ simplifies to $36^{\frac{1}{6}}$.